Let $G$ be a discrete countable group acting on a compact, Hausdorff space $X$. Assume that the action is transitive. Namely, $G\cdot x=X$, for all $x\in X$.
Does it follow that $X$ is finite?
I thought that the answer should be yes, as $G\cdot x$ is then a discrete compact space, and so must be finite. However, I am not sure anymore that I can claim that it is a discrete space. I'd appreciate any help.